1110 - MATH 153 Selected Solutions for § 11.10 Exercise 10...

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Unformatted text preview: MATH 153 Selected Solutions for § 11.10 Exercise 10 : Find the Maclaurin series for f ( x ) using the definition of a Maclaurin series. Also find the associated radius of convergence. f ( x ) = xe x Solution : We find a few derivatives to find the pattern in f ( n ) (0): f ( x ) = xe x f (0) = 0 f ( x ) = xe x + e x f (1) (0) = 1 f 00 ( x ) = xe x + 2 e x f (2) (0) = 2 f 000 ( x ) = xe x + 3 e x f (3) (0) = 3 It seems clear that f ( n ) ( x ) = xe x + ne x , so that f ( n ) (0) = n . Thus, xe x = ∞ X n =0 f ( n ) (0) n ! x n = ∞ X n =0 n n ! x n = ∞ X n =0 1 ( n- 1)! x n . To find the radius, we apply the Ratio Test: lim n →∞ x n +1 n ! x n ( n- 1)! = lim n →∞ | x | n +1 n ! · ( n- 1)! | x | n = lim n →∞ | x | n = 0 < 1 . Since the limit is always less than one regardless of x , the radius R = ∞ . Exercise 20 : Find the Taylor series for f ( x ) centered at the given value of a ....
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This note was uploaded on 07/26/2011 for the course MATH 153 taught by Professor Rempe during the Spring '08 term at Ohio State.

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1110 - MATH 153 Selected Solutions for § 11.10 Exercise 10...

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